Global well-posedness and blow-up of solutions for the Camassa–Holm equations with fractional dissipation

Gui, Guilong; Liu, Yue

doi:10.1007/s00209-015-1517-5

Cited by 11 publications

(4 citation statements)

References 26 publications

(42 reference statements)

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“…Afterwards, they proved the local wellposedness of the fractional nonlinear Schrödinger equation (R) in the subcritical Sobolev space H s (R) for s > n/2 − γ/2 in [11]. Gui and Liu [12] studied the Camassa-Holm equation with fractional dissipation, as follows:…”

Section: Introductionmentioning

confidence: 99%

On the Global Well-Posedness and Orbital Stability of Standing Waves for the Schrödinger Equation with Fractional Dissipation

2023

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In this paper, we are concerned with the nonlinear fractional Schrödinger equation. We extend the result of Guo and Huo and prove that the Cauchy problem of the nonlinear fractional Schrödinger equation is global well-posed in H32−γ(R) with 12≤γ<1. In view of the complexity of the nonlinear fractional Schrödinger equation itself, the local smoothing effect and maximal function estimates are not enough for presenting the global well-posedness for the nonlinear fractional Schrödinger equation. In this paper, we use a suitably iterative scheme and complete the global well-posed result for Equation (R). Moreover, we obtain the orbital stability of standing waves for the above equations via establishing the profile decomposition of bounded sequences in Hs(RN) (0<s<1) with N≥2.

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Section: Introductionmentioning

confidence: 99%

On the Global Well-Posedness and Orbital Stability of Standing Waves for the Schrödinger Equation with Fractional Dissipation

2023

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“…Recent years, increasing attention has been paid to the fractional equations. For instance, the fractional Korteweg-de Vries (fKdV) equation and the fractional Benjamin-Bona-Mahony (fBBM) equation have been obtained and studied in [18,29,37], the Camassa-Holm equations with fractional dissipation and the Camassa-Holm equations with fractional laplacian viscosity have been investigated in [24,25]. Concerning the fCH equation (1.1), the local well-posedness for initial data u 0 ∈ H s (R), s > 5 2 has been established by employing a semigroup approach due to Kato [31] in [34], which, to our knowledge, is the only result on the Cauchy problem for the fCH equation.…”

Section: Introductionmentioning

confidence: 99%

The Cauchy problem for fractional Camassa-Holm equation in Besov space

Fan

Gao

Wang

et al. 2020

Preprint

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In this paper, we consider the fractional Camassa-Holm equation modelling the propagation of small-but-finite amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. First, we establish the local well-posedness in Besov space B s 0 2,1 with s0 = 2ν − 1 2 for ν > 3 2 and s0 = 5 2 for 1 < ν ≤ 3 2 . Then, with a given analytic initial data, we establish the analyticity of the solutions in both variables, globally in space and locally in time.

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“…In stark contrast to the problem on the large time behavior for the Camassa-Holm equations without any nonlocal term (1.3), it seems fair to say that extremely little is known about the large time behavior for the solutions to the nonlocal equations (1.1)-(1.2)in two and three space dimensions. Indeed, to our best knowledge, the only example in [25] for which some results of the Camassa-Holm equations with fractional dissipation in one space dimension have been shown is the following:…”

Section: Introductionmentioning

confidence: 99%

“…In stark contrast to the problem on the regularity for the Camassa-Holm equations without any nonlocal term (1.3), it seems fair to say that extremely little is known about the regularity of the solutions to the nonlocal equations (1.1). Indeed, to our best knowledge, the only work for the nonlocal Camassa-Holm equations established by Gui-Liu [20], in which some results of the nonlocal Camassa-Holm equations in one space dimension have been obtained is as follows:…”

Section: Introductionmentioning

confidence: 99%

Large time behavior and convergence for the Camassa–Holm equations with fractional Laplacian viscosity

Gan

Meng

2018

Calc. Var.

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In this paper, we consider the n-dimensional (n = 2, 3) Camassa-Holm equations with fractional Laplacian viscosity in the whole space. In stark contrast to the Camassa-Holm equations without any nonlocal effect, to our best knowledge, little has been known on the large time behavior and convergence for the nonlocal equations under study. We first study the large time behavior of solutions. We then discuss the relation between the equations under consideration and the imcompressible Navier-Stokes equations with fractional Laplacian viscosity (INSF). The main difficulty to achieve them lies in the fractional Laplacian viscosity. Fortunately, by employing some properties of fractional Laplacian, in particular, the fractional Leibniz chain rule and the fractional Gagliardo-Nirenberg-Sobolev type estimates, the high and low frequency splitting method and the Fourier splitting method, we first establish the large time behavior concerning non-uniform decay and algebraic decay of solutions to the nonlocal equations under study. In particular, under the critical case s = n 4 , the nonlocal version of Ladyzhenskaya's inequality is skillfully used, and the smallness of initial data in several Sobolev spaces is required to gain the non-uniform decay and algebraic decay. On the other hand, by means of the fractional heat kernel estimates, we figure out the relation between the nonlocal equations under consideration and the equations (INSF). Specifically, we prove that the solution to the Camassa-Holm equations with nonlocal viscosity converges strongly as the filter parameter α → 0 to a solution of the equations (INSF).

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Global well-posedness and blow-up of solutions for the Camassa–Holm equations with fractional dissipation

Cited by 11 publications

References 26 publications

On the Global Well-Posedness and Orbital Stability of Standing Waves for the Schrödinger Equation with Fractional Dissipation

On the Global Well-Posedness and Orbital Stability of Standing Waves for the Schrödinger Equation with Fractional Dissipation

The Cauchy problem for fractional Camassa-Holm equation in Besov space

Large time behavior and convergence for the Camassa–Holm equations with fractional Laplacian viscosity

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